Twisted flag varieties of trialitarjan groups

For (A, σ) a central simple algebra of even degree with orthogonal involution, we present a method for constructing isotropic right ideals in the even Clifford algebra (C ₀(A, σ)σ) from isotropic right ideals in (A, σ). We then use this construction to fully describe the twisted flag varieties associated to algebraic groups of type D ₄ (including the trialitarian groups).     

R-equivalence and rationality problem for semisimple adjoint classical algebraic groups

The group of R-equivalence classes for all adjoint semisimple classical algebraic groups is computed. Examples of stably non-rational adjoint simple groups of type D _n ,n≥3, are presented. The complete stable birational classification of adjoint simple groups of rank 3 is given. This paper was written while the author was on sabbatical leave at the Université catholique de Louvain. Typeset byA _M S-T_EX     

Twisted loop groups and their affine flag varieties

We develop a theory of affine flag varieties and of their Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a “twisted case”; a consequence of our results is that our construction also includes the flag varieties for Kac–Moody Lie algebras of affine type. We also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k((t)) and some applications to local models of Shimura varieties.     

Twisted holomorphic forms on generalized flag varieties

In this paper we prove some vanishing theorems for the twisted Dolbeault cohomology of the complete flag varieties associated to a simple, simply connected algebraic group.     

Chow ring of generic flag varieties

Let G be a split semisimple algebraic group over a field k and let X be the flag variety (i.e., the variety of Borel subgroups) of G twisted by a generic G‐torsor. We start a systematic study of the conjecture, raised in 5 in form of a question, that the canonical epimorphism of the Chow ring of X onto the associated graded ring of the topological filtration on the Grothendieck ring of X is an isomorphism. Since the topological filtration in this case is known to coincide with the computable gamma filtration, this conjecture indicates a way to compute the Chow ring. We reduce its proof to the case of . For simply‐connected or adjoint G , we reduce the proof to the case of simple G . Finally, we provide a list of types of simple groups for which the conjecture holds. Besides of some classical types considered previously (namely, A, C, and the special orthogonal groups of types B and D), the list contains the exceptional types G₂, F₄, and simply‐connected E₆.     

Index reduction formulas for twisted flag varieties, I.

A general formula is proved for the change in the Schur index of a central simple algebra on passing from the ground field F to the function field F(X) of a twisted flag variety X, i.e., a projective variety such that there is an adjoint semisimple algebraic group G acting on X over F such that the action becomes transitive over the separable closure of F. The general formula encompasses special cases previously proved where X is a Brauer-Severi variety, or a generic partial splitting variety of a central simple algebra, or the transfer of such a variety, a quadric, or the involution variety of an algebra with orthogonal involution. For the classical simple groups G of inner type, all the corresponding varieties X are described, and the specific index reduction formula is given for each such X.The second author would like to express his thanks to J.-L. Colliot-Thélène for stimulating discussions on this subject.Supported in part by the NSF.     

Degree formula for connective K-theory

We use the degree formula for connective K-theory to study rational contractions of algebraic varieties. As an application we obtain a condition of rational incompressibility of algebraic varieties and a version of the index reduction formula. Examples include complete intersection, rationally connected varieties, twisted forms of abelian varieties and Calabi-Yau varieties     

Holomorphic K-Theory, Algebraic Co-cycles, and Loop Groups

In this paper we study the holomorphic K-theory of a projective variety. This K-theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory is built out of studying algebraic bundles over a variety up to algebraic equivalence. In this paper we will give calculations of this theory for flag like varieties which include projective spaces, Grassmannians, flag manifolds, and more general homogeneous spaces, and also give a complete calculation for symmetric products of projective spaces. Using the algebraic geometric definition of the Chern character studied by the authors we will show that there is a rational isomorphism of graded rings between holomorphic K-theory and the appropriate morphic cohomology groups, in terms of algebraic co-cycles in the variety. In so doing we describe a geometric model for rational morphic cohomology groups in terms of the homotopy type of the space of algebraic maps from the variety to the symmetrized loop group U(n)/ _n where the symmetric group _n acts on U(n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU(k) by a similar symmetric group action. We then use the Chern character isomorphism to prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K-theory by inverting the Bott class, then rationally this is isomorphic to topological K-theory. Finally this will allows us to produce explicit obstructions to periodicity in holomorphic K-theory, and show that these obstructions vanish for generalized flag mani-folds.     

Betti Numbers of Toric Varieties and Eulerian Polynomials

It is well known that the Eulerian polynomials, which count permutations in S _n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the S _n -orbit for a generic weight in the weight lattice of S _n . Therefore, the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra.

In this article we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a nongeneric weight, namely, a weight fixed by only the simple reflections J = {s _n , s _n?1, s _n?2,…, s _n?k+2, s _n?k+1} for some k with respect to the A _n root lattice. Furthermore, they give rise to certain rationally smooth toric varieties X(J) that come naturally from the theory of algebraic monoids. Using effectively the theory of reductive algebraic monoids and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincaré polynomial of X(J) in terms of the Eulerian polynomials.     

Algorithms for Computing Characters for Symmetric Spaces

Symmetric spaces or more general symmetric k-varieties can be defined as the homogeneous spaces G _k /K _k , where G is a reductive algebraic group defined over a field k of characteristic not 2, K the fixed point group of an involution θ of G and G _k resp. K _k the sets k-rational points of G resp. K. These symmetric spaces have a fine structure of root systems, characters, Weyl groups etc., similar to the underlying algebraic group G. The relationship between the fine structure of the symmetric space and the group plays an important role in the study of these symmetric spaces and their applications. To develop a computer algebra package for symmetric spaces one needs explicit formulas expressing the fine structure of the symmetric space and group in terms of each other. In this paper we consider the case that k is algebraically closed and give explicit algorithmic formulas for expressing the characters of the weight lattice of the symmetric space in terms of the characters of the weight lattice of the group. These algorithms can easily be implemented in a computer algebra package. The root system of the symmetric space can be described as the image of the root system of the group under a projection π derived from an involution θ on . This implies that . Using these formulas for the characters of each of these lattices we show that in fact . A.G. Helminck is partially supported by N.S.F. Grant DMS-0532140.     

Schubert cycles, differential forms and D-modules on varieties of unseparated flags

Proper homogeneous G-spaces (where G is semisimple algebraic group) over positive characteristic fields k can be divided into two classes, the first one being the flag varieties G/P and the second one consisting of varieties of unseparated flags (proper homogeneous spaces not isomorphic to flag varieties as algebraic varieties). In this paper we compute the Chow ring of varieites of unseparated flags, show that the Hodge cohomology of usual flag varieties extends to the general setting of proper homogeneous spaces and give an example showing (by geometric means) that the D -affinity of Beilinson and Bernstein fails for varieties of unseparated flags.     

Galois Stability, Integrality and Realization Fields for Representations of Finite Abelian Groups

For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups GGL _n (E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E=F(G) that are obtained via adjoining all matrix coefficients of all matrices gG to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E=F(G) for prescribed integers n and t or prescribed n and d.     

Twisted gamma filtration and algebras with orthogonal involution

For the Grothendieck group of a split simple linear algebraic group, the twisted γ-filtration provides a useful tool for constructing torsion elements in -rings of twisted flag varieties. In this paper, we construct a non-trivial torsion element in the γ-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline.     

Algorithms for Computations in Local Symmetric Spaces

In the last two decades much of the algebraic/combinatorial structure of Lie groups, Lie algebras, and their representations has been implemented in several excellent computer algebra packages, including LiE, GAP4, Chevie, Magma, and Maple. The structure of reductive symmetric spaces or more generally symmetric k-varieties is very similar to that of the underlying Lie group, with a few additional complications. A computer algebra package enabling one to do computations related to these symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc.

In this article we lay the groundwork for computing the fine structure of symmetric spaces over the real numbers and other base fields, give a complete set of algorithms for computing the fine structure of symmetric varieties and use this to compute nice bases for the local symmetric varieties.     

Homology stability for classical regular semisimple varieties

We use techniques from homotopy theory, in particular the connection between configuration spaces and iterated loop spaces, to give geometric explanations of stability results for the cohomology of the varieties of regular semisimple elements in the simple complex Lie algebras of classical type A, B or C, as well as in the group . We show that the cohomology spaces of stable versions of these varieties have an algebraic stucture, which identifies them as “free Poisson algebras” with suitable degree shifts. Using this, we are able to give explicit formulae for the corresponding Poincaré series, which lead to power series identities by comparison with earlier work. The cases of type B and C involve ideas from equivariant homotopy theory. Our results may be interpreted in terms of the actions of a Weyl group on its coinvariant algebra (i.e. the coordinate ring of the affine space on which it acts, modulo the invariants of positive degree; this space coincides with the cohomology ring of the flag variety of the associated Lie group) and on the cohomology of its associated complex discriminant variety. Received August 31, 1998; in final form August 1, 1999 / Published online October 30, 2000     

The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants

We give conditions on a curve class that guarantee the vanishing of the structure constants of the small quantum cohomology of partial flag varieties F(k ₁, ..., k _r ; n) for that class. We show that many of the structure constants of the quantum cohomology of flag varieties can be computed from the image of the evaluation morphism. In fact, we show that a certain class of these structure constants are equal to the ordinary intersection of Schubert cycles in a related flag variety. We obtain a positive, geometric rule for computing these invariants (see Coskun in A Littlewood–Richardson rule for partial flag varieties, preprint). Our study also reveals a remarkable periodicity property of the ordinary Schubert structure constants of partial flag varieties.     

Cohomologie des variétés de Deligne-Lusztig

We study the cohomology of Deligne-Lusztig varieties with aim the construction of actions of Hecke algebras on such cohomologies, as predicted by the conjectures of Broué, Malle and Michel ultimately aimed at providing an explicit version of the abelian defect conjecture. We develop the theory for varieties associated to elements of the braid monoid and partial compactifications of them. We are able to compute the cohomology of varieties associated to (possibly twisted) rank 2 groups and powers of the longest element w₀ (some indeterminacies remain for G₂). We use this to construct Hecke algebra actions on the cohomology of varieties associated to w₀ or its square, for groups of arbitrary rank. In the subsequent work [F. Digne, J. Michel, Endomorphisms of Deligne-Lusztig varieties, Nagoya J. Math. 183 (2006)], we construct actions associated to more general regular elements and we study their traces on cohomology.     

Algebraic quantum hypergroups

An algebraic quantum group is a regular multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We construct the dual, just as in the case of algebraic quantum groups and we show that the dual of the dual is the original quantum hypergroup. We define algebraic quantum hypergroups of compact type and discrete type and we show that these types are dual to each other. The algebraic quantum hypergroups of compact type are essentially the algebraic ingredients of the compact quantum hypergroups as introduced and studied (in an operator algebraic context) by Chapovsky and Vainerman.We will give some basic examples in order to illustrate different aspects of the theory. In a separate note, we will consider more special cases and more complicated examples. In particular, in that note, we will give a general construction procedure and show how known examples of these algebraic quantum hypergroups fit into this framework.     

The exponential image of simple complex lie groups of exceptional type

Let G be a connected reductive complex Lie group. Let E _G be the image of the exponential map of G and E' _G its complement in G. We give a purely algebraic characterization of the set E _G and also describe an algorithm for finding all conjugacy classes of G in E' _G . We are mainly interested in the case when the Lie algebra of G is simple and exceptional. Full details are provided for groups G of type G ₂, F ₄, and E ₆. If G is of type G ₂ then there are only two such conjugacy classes.This work was supported by NSERC Grant A-5285.